Brandon Roy scored eight of his 25 points in the final quarter as the Blazers turned a 68-64 deficit into a 79-68 lead.
Houston battled back to grab a four-point lead with 9:30 to play, but the Portland took the lead for good on a free throw by Steve Blake and three-pointers by Travis Outlaw and Roy with less than 5 minutes remaining.
"I've never questioned this team's heart," said Blazer Coach Nate McMillan. "Being in a situation like this, with your back against the wall, facing a very good team in Houston, they played well. They didn't want the season to end."
Roy and LaMarcus Aldridge both scored 25 for Portland while Blake added 15.
Luis Scola scored 21 to lead the Rockets, who will try to eliminate the Blazers in Game 6 Thursday in Houston. The Rockets are seeking to win their first playoff series since 1997 and end a string of six consecutive first-round exits.
"We just can't come down and play at a frenzied pace," said Houston Coach Rick Adelman. "We turned it over, we only had five assists in the second half. We just didn't play the way we needed to in order to win."
Portland led by as many as 10 points in the first quarter and led 50-43 at the half. Houston battled back in the third to tie the game 60-60 and trailed 64-62 going into the fourth quarter.